Positive solutions of second order elliptic equations in unbounded domains

Let $L$ be a second order uniformly elliptic operator defined in an unbounded domain $G$. The problem is considered of determining a solution positive in $G$ of the equation $Lu=0$ in $G$ with zero Dirichlet data on the boundary of the domain. The existence and uniqueness (to within multiplication by a positive constant) of a solution of this problem is established for some unbounded domains.
Bibliography: 3 titles.